ECE 754 Nonlinear Systems  Winter 2009 Instructor: Prof. Andrea Serrani. Office: 412 Dreese Lab.

Schedule: M-W-F 12:30-1:18pm
Room: 109 Caldwell Labs
Office hours: By appointment (email: serrani@ece.osu.edu) is definitely preferred, but feel free to knock at my door anytime.

Please note: The official web page of the course is developed on Carmen.
Registered students should access all the necessary information about the course (including notes, homework sets, solutions, and the updated syllabus) through Carmen.

Course Description
The course provides basic mathematical tools for the analysis of nonlinear dynamical systems, and introduces design techniques for the synthesis of nonlinear control systems. Topics include: properties of solutions of nonlinear differential equations, asymptotic behavior, Lyapunov stability theory, periodic orbits, linearization by feedback, control design by linearization methods, elementary local stabilization by state feedback. Additional topics (time permitting) may include elementary theory of dissipative systems, invariant manifold techniques, describing function method.

Course Goals (as stated in the ECE course listing)
1. Develop mathematical tools for analysis of nonlinear control systems.
2. Provide a thorough treatment of results from Lyapunov stability theory.
3. Introduce useful engineering concepts from numerical methods and phase-plane techniques.
4. Provide a basic treatment of design concepts for linearization via feedback.
5. Introduce examples and applications of nonlinear system modeling and control.

Prerequisites
State variables, differential equations, introductory linear systems theory, elementary stability theory (EE551 or graduate standing). Proficiency in basic engineering mathematics (e.g., linear algebra, multivariable calculus, and differential equations) is absolutely necessary. A basic, working knowledge of Matlab/Simulink is required. Use of mathematical software like Mathematica or Maple is encouraged, but not necessary.

Textbook and useful references:

• H.H Khalil, Nonlinear Systems (3rd edition), Prentice Hall, 2002.
  
  
• S.Sastry,  Nonlinear Systems. Analysis, Stability, and Control. Springer Verlag, 1999.
• M. Vidyasagar, Nonlinear Systems Analysis (2nd edition), Prentice Hall 1993.
• M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, 1974.
• W.J. Rugh, Linear Systems Theory, Prentice Hall, 1996.

 

Course outline

Introduction to nonlinear systems: an overview of peculiar behaviors of nonlinear systems vs. linear systems: multiple isolated equilibrium points, finite escape times, limit cycles.

Mathematical preliminaries: Quick review of normed vector spaces. Induced norms. Gronwall-Bellman inequality. Lipschitz condition.

Fundamental properties: Local and global existence and uniqueness of solutions. Continuity with respect to initial conditions. Finite escape times.

Phase plane analysis and local behavior of solutions near equilibria: Behavior of two-dimensional autonomous systems. Classification of equilibrium points. The Hartman-Grossman theorem. Periodic orbits. Bendixon's criterion. Index theory.

Geometric properties:  The flow of a vector field. Invariant sets. Asymptotic behavior of solutions: limit sets and the Poincare-Bendixon theorem.

Stability theory for autonomous systems: Stability of an equilibrium solution. Attractivity and uniform attractivity. Lyapunov stability. Global asymptotic stability: Barbashin-Krasovskii theorem. La Salle's invariance principle. Stability of linear systems: Lyapunov matrix equation. Stability by linearization.

Stability theory of non autonomous systems: Comparison functions. Uniform stability. Lyapunov theorems. Stability of linear time varying systems. Stability by linearization.

Introduction to systems with inputs:  Input-to-state stability. BIBO stability. Dissipative systems. Passive systems. Stabilization of passive systems.

Feedback linearization: Local relative degree. Lie derivatives. Lie bracket of vector fields. Invariant and involutive distributions. Zero-dynamics. Linearization by feedback. Input/output linearization. Sufficient conditions for  feedback stabilization.

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